If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.
Thanks to this, we know that △ ACD ~ △ ABE. Because of that, their corresponding side lengths are proportional.
AC/AB = AD/AE
Let's rewrite the numerators by using the Segment Addition Postulate.
AC = AB + BC
AD = AE + ED
Next, let's substitute the equations above into the proportion equation.
Given: & BE∥ CD
Prove: & ABBC = AEED
Proof: By the Corresponding Angles Theorem we get that ∠ ABE ≅ ∠ ACD and ∠ AEB ≅ ∠ ADC. Thus, △ ACD~△ ABE thanks to the AA Similarity Postulate. Then, we get the proportion ACAB = ADAE. The Segment Addition Postulates gives us AC = AB + BC and AD = AE + ED.
Substituting the last two equations into the proportion and performing the required operations, we get that ABBC = AEED, which is what we wanted.
